Related papers: Counting zero kernel pairs over a finite field
We prove that generating subspaces of matrix rings over finite fields are counted by polynomials. We use this result to define and study two-variable versions of polynomials counting isomorphism classes of absolutely irreducible…
About two decades ago, Tsfasman and Boguslavsky conjectured a formula for the maximum number of common zeros that $r$ linearly independent homogeneous polynomials of degree $d$ in $m+1$ variables with coefficients in a finite field with $q$…
Matrix completion aims to reconstruct a data matrix based on observations of a small number of its entries. Usually in matrix completion a single matrix is considered, which can be, for example, a rating matrix in recommendation system.…
Kernel quadratures and other kernel-based approximation methods typically suffer from prohibitive cubic time and quadratic space complexity in the number of function evaluations. The problem arises because a system of linear equations needs…
Criterion for a companion matrix to have a certain number of flat portions on the boundary of its numerical range is given. The criterion is specialized to the cases of 3-by-3 and 4-by-4 matrices. In the latter case, it is proved that a…
Let $R$ be a commutative ring and $M_n(R)$ be the ring of $n \times n$ matrices with entries from $R$. For each $S \subseteq M_n(R)$, we consider its (generalized) null ideal $N(S)$, which is the set of all polynomials $f$ with coefficients…
With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by…
We study the maximal number of pairwise distinct columns in a $\Delta$-modular integer matrix with $m$ rows. Recent results by Lee et al. provide an asymptotically tight upper bound of $O(m^2)$ for fixed $\Delta$. We complement this and…
An equivalence relation in the set of all square binary matrices is described in this work. It is discussed a combinatoric problem about finding the cardinal number and the elements of the factor set according to this relation. We examine…
The universality properties of kernels characterize the class of functions that can be approximated in the associated reproducing kernel Hilbert space and are of fundamental importance in the theoretical underpinning of kernel methods in…
Let $F$ be the finite field of order $q$ and $\M(n,r, F)$ be the set of $n\times n$ matrices of rank $r$ over the field $F$. For $\alpha\in F$ and $A\in \M(n,F)$, let $$Z^{\alpha}_{A,r}=\left\{X\in \M(n,r, F)\mid \tr(AX)=\alpha\right \}.$$…
This paper is concerned with the problem of determining the number of division algebras which share the same collection of finite splitting fields. As a corollary we are able to determine when two central division algebras may be…
Orthogonal and quasi-orthogonal matrices have a long history of use in digital image processing, digital and wireless communications, cryptography and many other areas of computer science and coding theory. The practical benefits of using…
This document is an exposition of an assortment of open problems arising from the exact enumeration of (perfect) matchings of finite graphs. Roughly half have been solved at the time of this writing; see the document "Twenty Open Problems…
Numerous results on self-reciprocal polynomials over finite fields have been studied. In this paper we generalize some of these to a-self reciprocal polynomials defined in [4]. We consider some properties of the divisibility of a-reciprocal…
The problem of classifying modules over a tame algebra A reduces to a block matrix problem of tame type whose indecomposable canonical matrices are zero- or one-parameter. Respectively, the set of nonisomorphic indecomposable modules of…
Two matrices are said non-overlapping if one of them can not be put on the other one in a way such that the corresponding entries coincide. We provide a set of non-overlapping binary matrices and a formula to enumerate it which involves the…
We find a formula for the number of permutation polynomials of degree q-2 over a finite field Fq, which has q elements, in terms of the permanent of a matrix. We write down an expression for the number of permutation polynomials of degree…
We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.
We discuss existence of explicit search bounds for zeros of polynomials with coefficients in a number field. Our main result is a theorem about the existence of polynomial zeros of small height over the field of algebraic numbers outside of…